T1 mapping is a quantitative magnetic resonance imaging (qMRI) technique that has emerged as a valuable tool in the diagnosis of diffuse myocardial diseases. However, prevailing approaches have relied heavily on breath-hold sequences to eliminate respiratory motion artifacts. This limitation hinders accessibility and effectiveness for patients who cannot tolerate breath-holding. Image registration can be used to enable free-breathing T1 mapping. Yet, inherent intensity differences between the different time points make the registration task challenging. We introduce PCMC-T1, a physically-constrained deep-learning model for motion correction in free-breathing T1 mapping. We incorporate the signal decay model into the network architecture to encourage physically-plausible deformations along the longitudinal relaxation axis. We compared PCMC-T1 to baseline deep-learning-based image registration approaches using a 5-fold experimental setup on a publicly available dataset of 210 patients. PCMC-T1 demonstrated superior model fitting quality (R2: 0.955) and achieved the highest clinical impact (clinical score: 3.93) compared to baseline methods (0.941, 0.946 and 3.34, 3.62 respectively). Anatomical alignment results were comparable (Dice score: 0.9835 vs. 0.984, 0.988). Our code and trained models are available at https://github.com/eyalhana/PCMC-T1.